Filomat 27:4 (2013), 679–691
DOI 10.2298/FIL1304679H
Published by Faculty of Sciences and Mathematics,
University of Ni
ˇ
s, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
Some Invariants of Quarter-Symmetric Metric Connections Under the
Projective Transformation
Yanling Han
a
, Ho Tal Yun
b
, Peibiao Zhao
c
a
Dept. of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094, P. R. China
School of Science, Qilu University of Technology, Jinan 250353, P. R. China
b
Faculty of Mathematics, Kim Tl Sung University, Pyongyang, Democratic People’s Republic of Korea
c
Dept. of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094, P. R. China
Abstract. This paper studies the characteristics of quarter-symmetric metric connections. Some invariants
with respect to the projective transformation are obtained.
1. Introduction
For the study of connection transformations, one of main topics is to consider the various manifolds
endowed with special connections, such as S. Fueki and Hiroshi Endo [10] investigated the contact metric
structure with Chern connection; I. E. Hirica [14] considered the pseudo-symmetric Riemannian space,
she indicated that any semi-symmetric manifold (R · R = 0) is of Ricci semi-symmetric (R · S = 0). M. M.
Tripathi [27] studied ξ−Ricci-semi-symmetric (κ, µ)−manifolds. Another topic is to consider some special
transformations corresponding to certain posed connections, for instance, N. S. Sinyukov [24] considered the
geodesic mapping of Riemannian spaces; P. Venzi [29] studied the celebrated geodesic mapping in pseudo-
Riemannian manifolds; J. V. Kiosak Mikeˇs and A. Vanˇzurov´a [16] also studied the geodesic mapping in
manifolds with affine connections; Liang [18] investigated the semi-symmetric connection; C. Udris¸te and
I. E. Hiric˘a [28] obtained the family of projective projections on tensors and connections; I. E. Hiricˇa and L.
Nicolescu [15] gave an algebraic characterization of the case when the conformal Weyl and conformal Lyra
connections have the same curvature tensor; G. Mˇargulescu in [19] studied the conformal transformation of
Minskowski spaces; F. Y. Fu, X. P. Yang and P. B. Zhao [9] consider a class of conformal mappings between two
semi-Riemannian manifolds and obtain the corresponding characteristics of geometries and physics for this
mapping. In particular, they proved that this type of conformal mapping keeps a generalized quasi-Einstein
manifold unchanged. Later, F. Y. Fu and P. B. Zhao [8] discussed the semi-symmetric projective mapping
in pseudo-symmetric Riemannian manifolds and proved that a semi-symmetric projective connection
mapping could change a pseudo-symmetric manifold (M; 1) into a locally pseudo-symmetric manifold.
As we know a linear connection
˜
∇ is symmetric if its torsion tensor
˜
T vanishes, otherwise it is non-
symmetric. We all know that a manifold with a symmetric linear connection is projectively flat if and only
2010 Mathematics Subject Classification. Primary 53C20; Secondary 53D11
Keywords. Connection Transformations, Semi-symmetric Connections, Weyl Curvature tensors, Quarter-symmetric Metric Con-
nections
Received: 5 March 2011; Accepted: 17 February 2013
Communicated by Vladimir Dragovic
Supported by a Grant-in-Aid for Science Research from Nanjing University of Science and Technology (2011YBXM120), by NUST
Research Funding No. CXZZ11-0258, AD20370 and by NNSF (11071119).
Email addresses: hanyanling1979@163.com (Yanling Han), hochong@163.com (Ho Tal Yun), pbzhao@njust.edu.cn (Peibiao Zhao)
Y.L. Han, H. T. Yun, P. B. Zhao / Filomat 27:4 (2013), 679–691 680
if the projective curvature tensor with respect to it vanishes identically. A linear connection
˜
∇ is a metric
connection if there is a Riemannian metric 1 in M such that
˜
∇1 = 0, otherwise it is non-metric. It is well
known that a linear connection
˜
∇ is symmetric and metric if and only if it is the Levi-Civita connection. In
1973, B. G. Schmidt [23] proved that if the holonomy group of
˜
∇ is a subgroup of the orthogonal group, then
˜
∇ is the Levi-Civita connection of a Riemannian metric. In 1932, H. A. Hayden [13] has induced the idea of
metric connection with torsion on a Riemannian manifold. Such a connection is called Hayden connection.
On the other hand, for a given 1-form λ in a Riemannian manifold, the Weyl connection constructed with
λ and its associated vector B [6] is a symmetric non-metric connection. In fact, the Riemannian metric of
the manifold is recurrent with respect to Weyl connection with the recurrence factor λ, that is,
˜
∇1 = λ ⊗ 1.
Another symmetric non-metric connection is projectively related to the Levi-Civita connection (see [3], [30]
for details).
A. Friedmann and J. A. Schouten [7] introduced the concept of the semi-symmetric linear connection
in a differential manifold in 1924. The linear connection
˜
∇ is said to be a semi-symmetric connection if its
torsion tensor
˜
T is of the form
˜
T(X, Y) = π(Y)X − π(X)Y, ∀X, Y ∈ χ(M),
where π is of 1-form associated with a vector P on M, and P is defined by 1(X, P) = π(X). In 1970, K.
Yano [30] considered a semi-symmetric metric connection (that means a linear connection is both metric
and semi-symmetric) on a Riemannian manifold and studied some of its properties. He proved that a
Riemannian manifold is conformally flat if and only if it admits a semi-symmetric metric connection whose
curvature tensor vanishes identically. He also proved that a Riemannian manifold is of constant curvature if
and only if it admits a semi-symmetric metric connection for which the manifold is a group manifold, where
a group manifold is a differential manifold admitting a linear connection
˜
∇ such that its curvature tensor
˜
R
vanishes and its torsion tensor
˜
T is covariantly constant with respect to
˜
∇. Liang in his paper [18] discussed
some properties of semi-symmetric metric connections and proved that the projective curvature tensor with
respect to semi-symmetric metric connections coincides with the projective curvature tensor with respect to
the Levi-Civita connection if and only if the characteristic vector is proportional to the Riemannian metric.
P. B. Zhao, H. Z. Song and X. P. Yang [38] introduced the concept of the projective semi-symmetric metric
connection, and found an invariant under the transformation of projective semi-symmetric connections
and indicated that this invariant could degenerate into the Weyl projective curvature tensor under certain
conditions, so the Weyl projective curvature tensor is an invariant as for the transformation of the special
projective semi-symmetric connection. To the study of semi-symmetric metric connections, the authors
propose other interesting results [32–34, 36, 37]. Recently, the authors in paper [35] even studied the theory
of transformations on Carnot Caratheodory spaces, and obtained the conformal invariants and projective
invariants on Carnot-Caratheodory spaces with the view of Felix Klein.
Then N. S. Agache and M. R. Chafle[1], U. C. De and S. C. Biswas [4] discussed a semi-symmetric non-
metric connection on a Riemannian manifold. If the semi-symmetric connection
˜
∇ satisfies the condition:
for any X, Y, Z ∈ χ(M), there holds
˜
∇
X
Y = ∇
X
Y + π(Y)X + 1(X, Y)P,
˜
∇
Z
1(X, Y) = −2π(X)1(Y, Z) − 2π(Y ) 1(X, Z),
where ∇ is the Levi-Civita connection, then
˜
∇ is called the semi-symmetric non-metric connection. This was
further developed by Agashe and Chafle [1], U. C. De and Kamily [5]. Agashe and Chafle [1] defined the
curvature tensor with respect to semi-symmetric non-metric connections. They proved the Weyl projective
curvature tensor with respect to semi-symmetric non-metric connections is equal to the Weyl projective
curvature tensor with respect to Riemannian connection, then they derived that a Riemannian manifold
with vanishing Ricci tensor with respect to semi-symmetric non-metric connections was projectively flat if
and only if the curvature tensor with respect to this semi-symmetric non-metric connection was vanished.
U. C. De and S. C. Biswas [4] discussed the semi-symmetric non-metric connection on Riemannian manifolds
by using these concepts and similar approaches, they studied some properties of the curvature tensor with
Y.L. Han, H. T. Yun, P. B. Zhao / Filomat 27:4 (2013), 679–691 681
respect to the semi-symmetric non-metric connection and proved that two semi-symmetric non-metric
connections would be equal under certain conditions.
The notion of the pseudo-symmetry [17] is a natural generalization of the semi-symmetric geometry [2]
along the line of spaces of constant sectional curvatures and locally symmetric spaces as follows:
R
0
⊂ R
1
⊂ R
2
⊂ R
3
,
where R
0
is the class of constant sectional curvature Riemann spaces, R
1
is the class of locally symmetric
Riemann spaces (i.e. ∇R = 0), R
2
is the class of semi-symmetric Riemann spaces (i.e. R·R = 0), R
3
is the class
of pseudo-symmetric Riemann spaces (i.e. R · R = LQ(1; R)). The class R
2
of semi-symmetric spaces was
introduced by E. Cartan, where R
2
-spaces were classified by Szab´o [26]. It is trivial that all semi-symmetric
manifolds are Ricci-semisymmetric (R · S = 0), but, in general, the converse criterion is not true unless the
Ricci semi-symmetric hypersurfaces of Euclidean spaces (n > 3) have positive scalar curvatures. However,
the problem that whether these notions are equivalent for hyper-surfaces in Euclidean spaces is still open.
In 1987, Gong [12] investigated the projective s-semi-symmetric connection on a Riemannian manifold
and proved that a Riemannian manifold was projectively flat if and only if there existed a s-semi-symmetric
connection with vanishing curvature tensors. He also proved that a Riemannian manifold M admitting s-
semi-symmetric connection was recurrent manifold, if the recurrence factor λ is closed, then M is projectively
flat.
In 1980, R. S. Mishra and S. N. Pardey [20] investigated quarter-symmetric metric connections in
Riemannian, Kaehlerian and Sasakian manifolds, they, in particular, studied Ricci quarter-symmetric metric
connections and obtained some properties of curvature tensors of these connections. They proved that if an
Einstein manifold M admits a quarter-symmetric metric connection whose curvature tensor vanishes, then
M is projectively flat, they got an necessary and sufficient condition that an Einstein manifold M associated
with a quarter-symmetric metric connection is a group manifold, and so on. While S. Golab [11] derived
Schouten’s and Struik’s, by using the second Bianchi identity, curvature tensor with respect to quarter-
symmetric connections. In 1982, K. Yano and T. Imai [31] studied quarter-symmetric metric connections
and gave some examples of these connections. They applied quarter-symmetric metric connections into
Hermitian manifold and proved that the covariant derivative of almost complex structure tensor F
h
i
with
respect to the Levi-Civita connection coincides with that of F
h
i
with respect to quarter-symmetric metric
connections; they also proved that a Kaehlerian manifold endowed with the quarter-symmetric metric
connection is flat when the curvature tensor vanishes. For the study of various types of quarter-symmetric
metric connections and applications, one can also see [11, 20, 22] for details.
Taking into account that the quarter-symmetric metric connection is a natural generalization of a semi-
symmetric metric connection, we would ask whether we can consider the invariants of quarter-symmetric
metric connections under some connection transformations just as the case of semi-symmetric metric
connections. In fact, there were few results about quarter-symmetric metric connections because of its
formal complexity and computational difficulty.
In this paper, we first consider the general form of quarter-symmetric metric connections and find a
semi-symmetric metric connection is indeed a special case with φ
j
i
= δ
j
i
; Then, we compute the curvature
tensor of quarter-symmetric metric connections, and study the properties of the projective transformation,
and give a sufficient condition that a linear connection is exactly a projective transformation of quarter-
symmetric metric connections, and find out some invariants under this connection transformation; At
last, we define and discuss the mutual connection of quarter-symmetric connections just as the mutual
connection of semi-symmetric connections and find the condition, under the connection transformation,
that keeps the curvature tensor unchanged.
The organization of this paper is as follows. In section 2, we will recall and give some necessary notations
and terminologies. Section 3 is devoted to the main theorems and their proofs. Some examples will appear
in the fourth section.
Y.L. Han, H. T. Yun, P. B. Zhao / Filomat 27:4 (2013), 679–691 682
2. Preliminaries
Let (M
n
, 1), n = dimM > 2, be an n-dimensional Riemannian manifold equipped with a Riemannian
metric 1, and ∇ be the Levi-Civita connection associated with 1. Let χ(M) denote the set of all tangent vector
fields on M.
Let D be a linear connection on M, if it satisfies
(D
X
1)(Y, Z) = 0, ∀X, Y ∈ χ(M), (2.1)
then D is called a metric connection.
We define the torsion tensor T of D by
T(X, Y) = ∇
X
Y − ∇
Y
X − [X, Y], ∀X, Y, Z ∈ χ(M). (2.2)
A metric connection D is called a quarter-symmetric metric connection if there holds
T(X, Y) = φ(X)π(Y) − φ(Y)π(X), ∀X, Y ∈ χ(M), (2.3)
where φ is a tensor field of type (1, 1), and π is a 1-form, called an associated 1-form.
Taking a local coordinate system in M such that 1, ∇, D, π , φ, T have the local expression, respectively,
1
ji
, {
h
ji
}, Γ
h
ji
, π
i
, φ
h
j
, T
h
ji
, then, by a direct computation, we have
T
h
ji
= π
j
φ
h
i
− π
i
φ
h
j
. (2.4)
Theorem 2.1. For a quarter-symmetric metric connection, in a local coordinate, there holds
Γ
h
ji
= {
h
ji
} +
1
2
π
j
(φ
ki
+ φ
ik
)1
kh
−
1
2
π
i
(φ
jk
− φ
kj
)1
kh
−
1
2
π
h
(φ
ji
+ φ
ij
), (2.5)
where π
j
= π
i
1
ij
.
Proof. Since D is a metric connection, then, ∀X, Y, Z ∈ χ(M), we have
(D
X
1)(Y, Z) = X(1(Y, Z)) − 1(D
X
Y, Z) − 1(Y, D
X
Z) = 0, (2.6)
(D
Y
1)(Z, X) = Y(1(Z, X)) − 1(D
Y
Z, X) − 1(Z, D
Y
X) = 0, (2.7)
(D
Z
1)(X, Y) = Z(1(X, Y)) − 1(D
Z
X, Y) − 1(X, D
Z
Y) = 0. (2.8)
By using (2.6), (2.7) and (2.8), we get
21(D
X
Y, Z) = 21(∇
X
Y, Z) + π(Y){1(φ(Z), X) + 1(φ(X), Z)}
− π(X){1(φ(Y), Z) − 1(φ(Z), Y)}
− π(Z){1(φ(X), Y) + 1(φ(Y), X)}. (2.9)
In a local coordinate (U, x
i
), we can choose
X =
∂
∂x
i
, Y =
∂
∂x
j
, Z =
∂
∂x
k
. (2.10)
Substituting (2.10) into (2.9) above, we have locally the following
2Γ
h
ji
1
hk
= 2{
h
ji
}1
hk
+ π
j
(φ
l
k
1
li
+ φ
l
i
1
lk
) − π
i
(φ
l
j
1
li
− φ
l
k
1
lj
) − π
k
(φ
l
i
1
lj
+ φ
l
j
1
li
)
= 2{
h
ji
}1
hk
+ π
j
(φ
ki
+ φ
ik
) − π
i
(φ
jk
− φ
kj
) − π
k
(φ
ij
+ φ
ji
), (2.11)
where φ
ij
= φ
h
i
1
hj
.
Contracting the above equality (2.11) with 1
kp
, then we get
2Γ
h
ji
δ
p
h
= 2{
k
ji
}δ
p
h
+ π
j
(φ
ki
+ φ
ik
)1
kp
− π
i
(φ
jk
− φ
kj
)1
kp
− π
p
(φ
ij
+ φ
ji
), (2.12)
where π
p
= π
k
1
kp
. Thus, we know that the equation (2.5) is tenable. This ends the proof of Theorem 2.1.
Y.L. Han, H. T. Yun, P. B. Zhao / Filomat 27:4 (2013), 679–691 683
Remark 2.2. We write
U
ij
=
1
2
(φ
ij
+ φ
ij
), V
ij
=
1
2
(φ
ij
− φ
ij
), (2.13)
then it is obvious that there exists the following
U
ij
= U
ji
, V
ij
= −V
ji
, (2.14)
which means V
ij
, U
ij
are of symmetric and of skew-symmetric, respectively, with respect to i, j, and
U
ij
+ V
ij
= U
ij
− V
ij
= φ
ij
.
Equation (2.5) is equivalent to
Γ
h
ji
= {
h
ji
} + π
j
U
h
i
− π
i
V
h
j
− π
h
U
ij
, (2.15)
where U
j
i
= U
ik
1
kj
, V
j
i
= V
ik
1
kj
.
Remark 2.3. If φ
j
i
is proportional to the identity tensor δ
j
i
, then the quarter-symmetric metric connection is reduced
into a semi-symmetric connection, and the coefficient is given as
Γ
h
ji
= {
h
ji
} + π
j
δ
h
i
− π
i
δ
h
j
− π
h
1
ji
.
3. Main Theorems and Proofs
Using (2.15) and the identity
R
h
kji
=
∂Γ
h
ji
∂x
k
−
∂Γ
h
ki
∂x
j
+ Γ
α
ji
Γ
h
kα
− Γ
α
ki
Γ
h
jα
. (3.1)
By a straightforward calculation, we find
R
h
kji
= K
h
kji
− U
h
k
π
ji
+ U
h
j
π
ki
− π
h
k
φ
ji
+ π
h
j
φ
ji
+ (∇
k
U
h
j
− ∇
j
U
h
k
)π
i
− (∇
k
U
ji
− ∇
j
U
ki
)π
h
− (∇
k
π
h
j
− ∇
j
π
h
k
)V
h
i
+ (π
j
∇
k
V
h
i
− π
k
∇
j
V
h
i
)
+ V
h
i
(U
t
k
π
j
− U
t
j
π
k
)π
i
− (π
k
U
ji
− π
j
U
ki
)V
t
i
π
h
− V
h
i
(U
h
k
π
j
− U
h
j
π
k
)V
t
i
π
t
+ (π
k
U
ji
− π
j
U
ki
)V
h
t
π
t
,
where K
h
kji
is the Riemanninan curvature tensor of ∇, π
j
i
= π
ih
1
hj
, and
π
ji
= ∇
j
π
i
− U
jα
π
α
π
i
+
1
2
π
α
π
α
U
ji
, (3.2)
where π
i
is the contravariant component of π
j
.
Now, let φ
ij
be symmetric, that is, V
ij
= 0, then the curvature tensor of the quarter-symmetric metric
connection D becomes
R
h
kji
= K
h
kji
− U
h
k
π
ji
+ U
h
j
π
ki
− π
h
k
φ
ji
+ π
h
j
φ
ji
+(∇
k
U
h
j
− ∇
j
U
h
k
)π
i
− (∇
k
U
ji
− ∇
j
U
ki
)π
h
.